(a) A continuous random variable $x$ has a probability density function defined
[tex]f(x)=\left\{\begin{array}{cc}
k\left(x-\frac{1}{a}\right), & 0 \leq x \leq 3 \\
0, & \text { elsewhere }\\
\end{array}\right.[/tex]
Given that [tex]P ( x \geq 1)=0.8[/tex], determine the:
(i) values of the constants a and k ;
(ii) mean of $x$.
Answer (2)
The values of the constants are a = − 1 and k = 15 2 . The mean of x is 1.8 .
;
The values of the constants are:
a = − 1
k = 15 2
The mean of x is:
1.8
Examples
Understanding probability density functions and calculating means is crucial in various fields. For instance, in finance, it helps in modeling stock prices and predicting potential returns. Imagine you're analyzing the daily price fluctuations of a stock. By fitting a probability density function to the historical price changes, you can estimate the likelihood of the stock reaching a certain price level. The mean of this distribution gives you an idea of the average expected price change, which is vital for making informed investment decisions. This problem showcases the fundamental steps in analyzing such distributions to extract meaningful insights.
